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Lab Rat guide to Fourier Transformations

Protein crystallography is one of those mysterious things that I always feel I should know more about. It starts with protein crystals, fair enough, and then heads rapidly out beyond the Magical Event Horizon leaving me with a picture of set of small fuzzy dots which mysteriously resolve themselves into an electron density map. Somewhere in all this someone will mention the dreaded Fourier transformation, at which point I know all hope is lost and a might as well stop listening.

With the help of a good lecture and an utterly outstanding website though, I'm starting to get the hang of it. X-ray crystallography works by shooting X-rays at a protein crystal. As the crystal is in a regular lattice structure, the X-rays are scattered in a regular way. Each scattered ray can be characterised by the amplitude and the phase, as shown below:Each dot on the resulting image (which usually looks something like the figure shown on the right) corresponds to a scattered X-ray. The position of the dot gives (by way of a set of clever equations) the amplitude of the wavelength. However in order to calculate the position of an atom you need both the amplitude and the phase, and you can't get the phase from just the position of the dot. And here is where the dreaded Fourier transformation comes in, to help give information about the phase.

The Fourier transformation is an equation, which gives reciprocal information about a molecule, for example if the molecule is a single small point, the Fourier transformation (i.e what it looks like after plugging its positional coordinates into the Fourier equation) will be a large fuzzy blob, as shown below (all pictures from now on are taken - with permission - from Kevin Cowtan's Book of Fourier) . Molecule is on the left and its Fourier transformation on the right:For the more mathematically minded, what the Fourier transformation actually does is take a function and express it as the sum of a set of different sine and cosine waves. Apparently this can be done for any continuous function. By using this transformation you can get not only the amplitude, but also the phase of any given molecule inside an atom. You can also make the equation go backwards as well, turning the red fuzzy blob on the left back into the individual point. As most molecules contain many, many atoms there are various tricks you have to do in order to make this easier for large molecules, but first a quick proof of this concept using my favourite pictures on the Cowtan website.

Here is a picture of a duck:And here is the Fourier transformation of the duck, with the amplitude represented by the brightness of the colour and the phase represented by the actual colour:In order to show that the Fourier transformation really does show the phase, this transformation is mixed with the Fourier transformation of a picture of a cat. Rather than mix them both equally, the amplitude (brightness) of the duck transformation is mixed with the phase (actual colour) of the cat to give the following Fourier transformation:Performing the Fourier equation on this blob (to turn it back into an animal again) gives the following result, hopefully not unsurprising if I've managed to explain this alright:It's a cat! A rather blotchy cat, true, with a fairly trippy background, but nevertheless the Fourier transformation has faithfully reproduced the phases, rather than the amplitudes.

So how does this help when looking at molecules? It turns out that the fuzzy black-and-gray-dots picture that is the end result of X-ray crystallography (shown above and reproduced here on the right) is the Fourier transform of the atomic electron clouds inside the protein crystal. That picture is like the fuzzy coloured blobs that the duck and cat images came out of. In the same way that those turned into animals, this picture can be turned into the approximate shape of the electron clouds surrounding a molecule. For low resolution images this can show the secondary structure of a protein, the positions of alpha helices and beta sheets and a general idea of protein shape. For high resolution images, individual amino-acid residues can be seen, allowing a much more detailed view of the structure to be generated.

It isn't always perfect. Sometimes you do get the equivalent of a blobby cat with a trippy background and have to play around with homologous comparisons and allowed bond-angles to get a meaningful structure. There are plenty of strategies that exist to help you get a better image as well, particuarly for larger molecules which need more help resolving phases. From what I've heard though, once you've actually got the crystal, the rest seems like childs play in comparison. I know people who have spent their whole PhD's, and longer, just trying to isolate and concentrate a single protein crystal...

The axis are I and R. For slightly more clarification on the colours, positive real numbers are red, and negative real numbers are cyan. White represents zero magnitude.

To be honest the actual 'mathematics' of the Fourier transform is still a little beyond me, probably because I've never been formally taught about imaginary numbers. But I'm pleased that I've at least got a vague idea of how it helps with crystallography kind of sorted in my head.

Nice summary, but gosh, don't go saying that it's all straight forward once you get a crystal, lol. It really does come down to the luck of the draw. It should be straight-forward, but there are still hurdles.

It can certainly take a while to get a crystal (it took me three years), but that setting up screening trays by hand. This is old school now as the high through-put approaches can do 10,000 conditions a day using a fraction of a ul of protein - plus cameras can recognise crystalline birefringency and modify the pH, salt conc and precipitant concentration around that hit.

You may find that a seemingly good crystal offers very poor resolution, i.e. >4 A. Although the improvement in data crunching software and processor power mean that you can derive more structural information from poor resolutions that you once could, especially if you can model it onto a related known structure, or if you've used a phasing technique such as substituting selenomethionines for methionine.

However, one nightmare situation you might encounter, even with a beautiful, large crystal, is finding yourself sat at the ESRF in Grenoble only to discover from the initial cursory data analysis that your crystal is 40%+ twinned.

Depending on the form of twinning, you may find that despite a resolution of 2 A, the fantastic data from one crystal space-group is wiped out by the data from the twinning space group crystal.

My experiance of crystallography comes from hearing the experiances of various friends complaining about how difficult getting a workable crystal was. As such, I think my view of crystallography is probably a little skewed!

That does sound like a painful experiance. Should have known: very little biology is ever as simple as it sounds on paper...

William and Lawrence Bragg discovered X-ray crystallography by finding the structure of a salt (NaCl) crystal. They were from Adelaide, where I am, and then moved to London to finish the research. I watched a doco on them just the other day actually! Interesting stuff, but a bit too close to physics for my liking!